I cannot speak for all the wealthy and well-connected Jews in Nazi-occupied countries in general. But there are some people who were both deemed Jewish and managed to escape genocide through wealth and fame.
The famous Austrian philosopher, Ludwig Wittgenstein, was considered by the Reich to be Jewish. So was his family. At the time Ludwig Wittgenstein was in Britain and was able to obtain a lecturing post at Cambridge University. Later on, he was able to obtain British citizenship.
Many of his family members still resided in Austria, and were not quite as fortunate. They tried convincing the Nazi authorities that the Wittgensteins were Aryans. One family member tried to convince Nazi authorities that the grandfather of Ludwig Wittgenstein, Herman Christian Wittgenstein, was the illegitimate son of a non-Jewish prince. They bought Yugoslav passports in an unsuccessful attempt to escape to Switzerland; they were arrested but ultimately not convicted. (According to Ludwig Wittgenstein's sister, Hermine, it was principally their status as rich and famous Austrians that got all the charges dropped.) The Reich was not convinced by these appeals, which rested on ideological grounds. But they were persuaded on pragmatic grounds. You see, the Wittgenstein family was loaded. And the Nazis wanted their money.
The Reichsbank was authorized to seize the money, but for bureaucratic reasons that I am not totally clear on, they had difficulty withdrawing the Wittgenstein family's money from Switzerland. Gretl Wittgenstein and Ludwig Wittgenstein (who by then were American and British citizens, respectively) used this as a negotiation tactic: count Hermine and Helene Wittgenstein as Aryans, and they would hand over the money. The end result was that most of the Wittgenstein fortune was sent to the Reichsbank and the spurious claim of Herman Christian Wittgenstein's princehood was accepted. In August 1939, all grandchildren of Herman Christian Wittgenstein were considered to be of mixed Jewish blood. In February 1940, they were granted exceptional status as Aryans. And so they managed to sit out the entire Nazi occupation, including the Holocaust, mostly unbothered.
P.S. Incidentally, the famous psychoanalyst, Sigmund Freud, was allowed to depart Austria thanks to international pressure. In exchange, he had to sign a statement saying that he was treated well by the Nazis. But he died before the Holocaust.
CITATIONS
Stephen Budiansky, Journey to the Edge of Reason.
Ray Monk, Ludwig Wittgenstein: The Duty of Genius.
This question, although many times implied in bibliography, has seldomly been addressed analytically for the loaded implications it has. Also it is objectively impossible to answer that question on a continent-wide basis, so I will limit myself to Greece which on the whole does not present as an exception.
First of all we need to define "wealth" and "well-connected". When we speak of "wealthy" Jews in Greece we do not only speak of bank account sizes but also of a certain educational level with wealthy Jews being able to speak not only the native Ladino but also Greek, (although there were native Greek speakers from pre-1912 Greece, the majority of the population were Ladino speakers from the post-1912 territories).
Also "well-connected" can be interpreted in two ways. One is well-connected as far as integration in the Greek (Christian) society fabric with business or family ties and connections with Greek (Christian) authority. But in the context of Greek Resistance it can also mean ties with the Greek Communist Party which dominated thoroughly this area.
The simple answer to this question is yes. For Jews to escape the Holocaust in Greece, they would either need to:
Each of these choices demanded Christian intermediaries who would share the risk for either ideological or pecuniary reasons reasons. But since each escape demanded the cooperation of different people, usually a combination of these factors was necessary.
Nearly all cases of Jews hiding, albeit a very limited number, was consisted of people with strong ties with the Christian population on a family level. Almost none was linked as to "how well connected" they were, despite that in cases as Salonica many Jews indeed were well connected. While people who helped them usually did it out of personal desire, these Jews to be able to have formed such ties they were usually middle to upper class. In all of these stories money plays a central role, although we should not prescribe pecuniary motives to those who helped them. It was simply impossible to survive in a city without money.
Jews hiding in the countryside always included the cooperation with the Resistance. In these cases while money was necessary, having more did not necessarily mean better chances of survival. In all testimonies money is used as a means to survive but very seldom the lack of it prevents people from continuing their escape. More important was the ability to prove useful to the rural population which provided a means of survival, eg a tinsmith, a doctor or a teacher.
Escaping to the Resistance needed connections but these were not hard-pressed to find. Both because of the extent of the Resistance but also because of an important Jewish involvement in the Communist party. In these cases hands ready to carry arms was valued, rather than money or personal connections.
Escape to the Middle East almost always was linked to people with connections big or small and some level of economic wealth. Because it included the cooperation of the Resistance, the Allies and Christian locals it was impossible not to find a link in the chain were money was absolutely necessary. There are documents from the Greek Resistance requesting material support from the Yishuv, although not necessarily linking it to continuing the help. But we also have many reports of evacuees being strip searched before boarding kaiques or even being denied transport because of lack of money.
So given that wealth did confer advantages to those trying to escape and "connections" were meaningful only on a personal level, we must admit that their socio-economic status did not matter on the grand scheme of things. Although in the past factors like the limited integration of Ladino-speaking Jews was advanced, this did not explain how the Romaniotes faced similar fate. The reality is that the principal factor were the Germans themselves; in Salonica they had already perfected their modus operandi and was the first city to strike. There poor or rich suffered the same fate, well-connected to Christian authorities faced a hostile Christian population and well-connected to the Resistance faced indifference.
In other cities, Salonica served as an example but also other external factors proved important. For example the communities of Katerini and Zante were saved, poor or rich, because of good relations with the local population but also lack of fervor by local Occupation commanders. This is corroborated by the less success, but still success, in other cities like Volos or the Greek hinterland were Jews were able to flee to the rural areas or the Resistance but were hunted by the Germans.
The same external reasons were the key in the complete annihilation of Jews in Eastern Macedonia, (the greek part of Macedonia, not the current North Macedonian state) and Thrace. These areas were occupied by Bulgaria, which was a junior Nazi ally. Although Bulgarians tout their opposition of deporting Bulgarian Jews, they were more than happy to deport to their extermination of Greek Jewry in those areas. Under the brutal, (Bulgaria aimed in annexing these territories and proceeded in a violent bulgarization) bulgarian occupation wealth was un-important and connections useless.
So when trying to answer your question we must do a bit of re-framing. Indeed as we saw socio-economic status did matter. But it was only after having taking into account the more important factors of when and where a Jew was. If the answer was early 1943 Salonica, then the answer was simply it did not matter. The same answer was also valid for Zante or bulgarian-held Xanthi, but for different reasons each.
Where it did matter was late 1943 Athens or other cities. If the Jew had skills or could carry arms then wealth was not necessary but it surely helped in providing flexibility, since it allowed for non-productive members of the family to follow him which was a main concern which appears in all testimonies of the survivors. On the other hand money was absolutely necessary to flee Greece and lack of it could become damning.
As a last tidbit of information I would like to add that after the war, the biggest division was between those who had returned from the camps and those who had escaped through the Resistance/hinterland. There does appear to be a correlation between the poorer "Poles" survivors of the lagers (on a sidenote this self-determination would be comically punishable by the current polish laws forbidding the term) and the more affluent "Mountain (people)" who survived by fleeing to the countryside. Although correlation does not mean causation, it does tend to corroborate that if all other more important factors were set aside, it did play a role. Still, despite internal strife which spared no harsh words, I have not located any kind of reference or accusation of buying their way to safety or the rich enjoying privileges that the poor didn't during the persecution.
Although I cannot speak about the rest of Europe, this analysis of Jewish means of escape does tend to fall along the results by other analysis who tried to link Jewish status or behaviour with survivor rates. I am specifically referring to the discourse on the Judenrats and if their behavior, cooperation/resistance/perfunctory, was in any way a factor in survivor rates; it wasn't with time within the Holocaust timeframe and external Christian populations attitudes making the Judenrat role meaningless in the final outcome.
Sources: "H διάσωση" by Iakov Schiby. "Συναγωνιστές" by Iassonas Chandrinos "Ντοκουμέντα" by Natan Greenberg and in general work by V.Ritzaleos "The Holocaust in Thessaloniki, Reactions to Anti_Jewish persecution" by Leon Saltiel "Jewish Resistance in wartime Greece" by Steven Bowman